Bit mapping scheme for an ldpc coded 16apsk system

ABSTRACT

A digital communication system, having a transmitter to transmit a digital signal; and a receiver to receive the digital signal; wherein the digital signal utilizes a 16APSK system, and the signal is bit-mapped using gray mapping, and bits of the digital signal are ordered based on the values of a log likelihood ratio from a communications channel.

RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.11/813,203, filed Jun. 29, 2007, which is the U.S. National Stage ofInternational Application No, PCT/CN2006/002423, filed Sep. 18, 2006 andclaims the benefit thereof. This Application relates to application Ser.No. 11/813,208, filed Jun. 29, 2007, and application Ser. No.11/813,201, filed Jun. 29, 2007.

FIELD OF THE INVENTION

The invention relates to digital communications and in particular to abit mapping scheme for an LDPC coded 16APSK System.

BACKGROUND OF THE INVENTION

Forward Error Control (FEC) coding is used by communications systems toensure reliable transmission of data across noisy communicationchannels. Based on Shannon's theory, these communication channelsexhibit a fixed capacity that can be expressed in terms of bits persymbol at a given Signal to Noise Ratio (SNR), which is defined as theShannon limit. One of the research areas in communication and codingtheory involves devising coding schemes offering performance approachingthe Shannon limit while maintaining a reasonable complexity. It has beenshown that LDPC codes using Belief Propagation (BP) decoding provideperformance close to the Shannon limit with tractable encoding anddecoding complexity.

In a recent paper Yan Li and William Ryan, “Bit-Reliability Mapping inLDPC-Codes Modulation systems”, IEEE Communications Letters, vol. 9, no.1, January 2005, the authors studied the performance of LDPC-codedmodulation systems with 8PSK. With the authors' proposed bit reliabilitymapping strategy, about 0.15 dB performance improvement over thenon-interleaving scheme is achieved. Also the authors show that graymapping is more suitable for high order modulation than other mappingscheme such as natural mapping.

BRIEF SUMMARY OF THE INVENTION

Various embodiments of the present invention are directed to a bitmapping scheme in a 16APSK modulation system. The techniques of theseembodiments are particularly well suited for use with LDPC codes.

LDPC codes were first described by Gallager in the 1960s. LDPC codesperform remarkably close to the Shannon limit. A binary (N, K) LDPCcode, with a code length N and dimension K, is defined by a parity checkmatrix H of (N-K) rows and N columns. Most entries of the matrix H arezeros and only a small number the entries are ones, hence the matrix His sparse. Each row of the matrix H represents a check sum, and eachcolumn represents a variable, e.g., a bit or symbol. The LDPC codesdescribed by Gallager are regular, i.e., the parity check matrix H hasconstant-weight rows and columns.

Regular LDPC codes can be extended to form irregular LDPC codes, inwhich the weight of rows and columns vary. An irregular LDPC code isspecified by degree distribution polynomials v(x) and c(x), which definethe variable and check node degree distributions, respectively. Morespecifically, the irregular LDPC codes may be defined as follows:

$\begin{matrix}{{{v(x)} = {\sum\limits_{j = 1}^{d_{v\; \max}}{v_{j}x^{j - 1}}}},{and}} & (1) \\{{{c(x)} = {\sum\limits_{j = 1}^{d_{c\max}}{c_{j}x^{j - 1}}}},} & (2)\end{matrix}$

where the variables d_(v max) and d_(c max) are a maximum variable nodedegree and a check node degree, respectively, and v_(j) (c_(j))represents the fraction of edges emanating from variable (check) nodesof degree f. While irregular LDPC codes can be more complicated torepresent and/or implement than regular LDPC codes, it has been shown,both theoretically and empirically, that irregular LDPC codes withproperly selected degree distributions outperform regular LDPC codes.FIG. 1 illustrates a parity check matrix representation of an exemplaryirregular LDPC code of codeword length six.

LDPC codes can also be represented by bipartite graphs, or Tannergraphs. In Tanner graph, one set of nodes called variable nodes (or bitnodes) corresponds to the bits of the codeword and the other set ofnodes called constraints nodes (or check nodes) corresponds the set ofparity check constrains which define the LDPC code. Bit nodes and checknodes are connected by edges, and a bit node and a check node are saidto be neighbors or adjacent if they are connected by an edge. Generally,it is assumed that a pair of nodes is connected by at most one edge.

FIG. 2 illustrates a bipartite graph representation of the irregularLDPC code illustrated in FIG. 1.

LDPC codes can be decoded in various ways such as majority-logicdecoding and iterative decoding. Because of the structures of theirparity cheek matrices, LDPC codes are majority-logic decodable. Althoughmajority-logic decoding requires the least complexity and achievesreasonably good error performance for decoding some types of LDPC codeswith relatively high column weights in their parity check matrices(e.g., Euclidean geometry LDPC and projective geometry LDPC codes),iterative decoding methods have received more attention due to theirbetter performance versus complexity tradeoffs. Unlike majority-logicdecoding, iterative decoding processes the received symbols recursivelyto improve the reliability of each symbol based on constraints thatspecify the code. In a first iteration, an iterative decoder only uses achannel output as input, and generates reliability output for eachsymbol.

Subsequently, the output reliability measures of the decoded symbols atthe end of each decoding iteration are used as inputs for the nextiteration. The decoding process continues until a stopping condition issatisfied, after which final decisions are made based on the outputreliability measures of the decoded symbols from the last iteration.According to the different properties of reliability measures usedduring each iteration, iterative decoding algorithms can be furtherdivided into hard decision, soft decision and hybrid decisionalgorithms. The corresponding popular algorithms are iterativebit-flipping (BF), belief propagation (BP), and weighted bit-flipping(WBF) decoding, respectively. Since BP algorithms have been proven toprovide maximum likelihood decoding when the underlying Tanner graph isacyclic, they have become the most popular decoding methods.

BP for LDPC codes is a type of message passing decoding. Messagestransmitted along the edges of a graph are log-likelihood ratio

$({LLR})\log \frac{p_{0}}{p_{1}}$

associatea with variable nodes corresponding to codeword bits. In thisexpression p₀ and p₁ denote the probability that the associated bitvalue becomes either a 0 or a 1, respectively. BP decoding generallyincludes two steps, a horizontal step and a vertical step. In thehorizontal step, each check node c_(m) sends to each adjacent bit b_(n)a check-to-bit message which is calculated based on all bit-to-checkmessages incoming to the check c_(m) except one from bit b_(n). In thevertical step, each bit node b_(n) sends to each adjacent check nodec_(m) a bit-to-check message which is calculated based on allcheck-to-bit messages incoming to the bit b_(n) except one from checknode c_(m). These two steps are repeated until a valid codeword is foundor the maximum number of iterations is reached.

Because of its remarkable performance with BP decoding, irregular LDPCcodes arc among the best for many applications. Various irregular LDPCcodes have been accepted or being considered for various communicationand storage standards, such as DVB-S2/DAB, wireline ADSL, IEEE 802.11n,and IEEE 802.16.

The threshold of an LDPC code is defined as the smallest SNR value atwhich, as the codeword length tends to infinity, the bit errorprobability can be made arbitrarily small. The value of threshold of anLDPC code can be determined by analytical tool called density evolution.

The concept of density evolution can also be traced back to Gallager'sresults. To determine the performance of BE decoding, Gallager derivedformulas to calculate the output BER for each iteration as a function ofthe input BER at the beginning of the iteration, and then iterativelycalculated the BER at a given iteration. For a continuous alphabet, thecalculation is more complex. The probability density functions (pdf's)of the belief messages exchanged between bit and check nodes need to becalculated from one iteration to the next, and the average BER for eachiteration can be derived based on these pdf's. In both check nodeprocessing and bit node processing, each outgoing belief message is afunction of incoming belief messages.

For a check node of degree d_(c), each outgoing message U can beexpressed by a function of d_(c)−1 incoming messages,

U=F _(c)(V ₁ , V ₂ , . . . , V _(d) _(c) ⁻¹).

where F_(c) denotes the check node processing function which isdetermined from BP decoding. Similarly, for bit node of degree d_(v),each outgoing message V can be expressed by a function of d_(v)−1incoming messages and the channel belief message U_(ch),

V=F _(V)(U _(ch) , U ₁ , U ₂ , . . . , U _(d) _(v) ⁻¹).

where F_(v) denotes the bit node processing function. Although for bothcheck and bit node processing, the pdf of an outgoing message can bederived based on the pdf's of incoming messages for a given decodingalgorithm, there may exist an exponentially large number of possibleformats of incoming messages. Therefore the process of density evolutionseems intractable. Fortunately, it has been proven in that for a givenmessage-passing algorithm and noisy channel, if some symmetry conditionsare satisfied, then the decoding BER is independent of the transmittedsequence x. That is to say, with the symmetry assumptions, the decodingBER of all-zero transmitted sequence x=1 is equal to that of anyrandomly chosen sequence, thus the derivation of density evolution canbe considerably simplified. The symmetry conditions required byefficient density evolution are channel symmetry, check node symmetry,and bit node symmetry, Another assumption for the density evolution isthat the Tanner graph is cyclic free.

According to these assumptions, the incoming messages to bit and checknodes are independent, and thus the derivation for the pdf of theoutgoing messages can be considerably simplified. For many LDPC codeswith practical interests, the corresponding Tanner graph containscycles. When the minimum length of a cycle (or girth) in a Tanner graphof an LDPC code is equal to 4×l, then the independence assumption doesnot hold after the l-th decoding iteration with the standard BPdecoding. However, for a given iteration number, as the code lengthincreases, the independence condition is satisfied for an increasingiteration number. Therefore, the density evolution predicts theasymptotic performance of an ensemble of LDPC codes and the “asymptotic”nature is in the sense of code length.

According to various embodiments of the invention, the bit mappingschemes provide good threshold of LDPC codes. Furthermore, the bitmapping schemes can facilitate the design of an interleaving arrangementin a 16APSK modulation system.

According to various embodiments of the invention, the disclosed bitmapping offers good performance of LDPC coded 16APSK system andsimplifies an interleaving arrangement in 16APSK systems.

According to various embodiments of the invention, a method of bitmapping in a 16APSK system, wherein the system utilizes an FEC code,comprises: transmitting a digital signal from a transmitter; andreceiving the digital signal at a receiver; wherein the digital signalutilizes a 16APSK system with FEC coding, and the signal is bit-mappedprior to the transmitting according to a formula embodied by FIG. 4.

According to various embodiments of the invention, the FEC code isregular LDPC code.

According to various embodiments of the invention, the FEC code isirregular LDPC code.

According to various embodiments of the invention, the FEC code isregular repeat-accumulate code.

According to various embodiments of the invention, the FEC code isirregular repeat-accumulate code.

According to various embodiments of the invention, a digitalcommunication system, comprises: a transmitter to transmit a digitalsignal; and a receiver to receive the digital signal; wherein thedigital signal utilizes a 16APSK system with FEC coding, and the signalis bit-mapped prior to the transmitting according to a formula embodiedby FIG. 4.

According to various embodiments of the invention, a digitalcommunication system comprises: a transmitter to transmit a digitalsignal; and a receiver to receive the digital signal; wherein thedigital signal utilizes a 16APSK system with FEC coding, and the signalis bit-mapped using gray mapping, and bits of the digital signal areordered based on the values of a log likelihood ratio from acommunications channel.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example, and not by wayof limitation, in the figures of the corresponding drawings and in whichlike reference numerals refer to similar elements and in which:

FIG. 1 is a parity check matrix representation of an exemplary irregularLDPC code of codeword length six.

FIG. 2 illustrates a bipartite graph representation of the irregularLDPC code illustrated in FIG. 1.

FIG. 3 illustrates the bit mapping block in 16APSK modulation, accordingto various embodiments of the invention,

FIG. 4 illustrates a bit map for 16APSK symbol, according to variousembodiments of the invention.

FIG. 5 depicts an example of a communications system which employs LDPCcodes and 16APSK modulation, according to various embodiments of theinvention.

FIG. 6 depicts an example of a transmitter employing 16APSK modulationin FIG. 5, according to various embodiments of the invention.

FIG. 7 depicts an example of a receiver employing 16APSK demodulation inFIG. 5, according to various embodiments of the invention.

DETAILED DESCRIPTION OF THE INVENTION

Referring to the accompanying drawings, a detailed description will begiven of exemplary encoded bit mapping methods using LDPC codesaccording to various embodiments of the invention,

Although the invention is described with respect to LDPC codes, it isrecognized that the bit mapping approach can be utilized with othercodes. Furthermore, it is recognized that this approach can beimplemented with uncoded systems.

FIG. 5 is an exemplary diagram of a communications system employing LDPCcodes with 16APSK modulation, according to various embodiments of thepresent invention. The exemplary communications system includes atransmitter 501 which generates signal waveforms across a communicationchannel 502 to a receiver 503. The transmitter 501 contains a messagesource for producing a discrete set of possible messages. Each of thesemessages corresponds to a signal waveform. The waveforms enter thechannel 502 and are corrupted by noise. LDPC codes are employed toreduce the disturbances introduced by the channel 502, and a 16APSKmodulation scheme is employed to transform LDPC encoded bits to signalwaveforms.

FIG. 6 depicts an exemplary transmitter in the communications system ofFIG. 5 which employs LDPC codes and 16APSK modulation. The LDPC encoder602 encodes information bits from source 601 into LDPC codewords. Themapping from each information block to each LDPC codeword is specifiedby the parity check matrix (or equivalently the generator matrix) of theLDPC code. The LDPC codeword is interleaved and modulated to signalwaveforms by the interleaver/modulator 603 based on a 16APSK bit mappingscheme. These signal waveforms are sent to a transmit antenna 604 andpropagated to a receiver shown in FIG. 7.

FIG. 7 depicts an exemplary receiver in FIG. 5 which employs LDPC codesand 16APSK demodulator. Signal waveforms are received by the receivingantenna 701 and distributed to demodulator/deinterleavor 702. Signalwaveforms are demodulated by demodulator and deinterleaved bydeinterleavor and then distributed to a LDPC decoder 703 whichiteratively decodes the received messages and output estimations of thetransmitted codeword. The 16APSK demodulation rule employed by thedemodulator/deinterleaver 702 should match with the 16APSK modulationrule employed by the interleaver/modulator 603.

According to various embodiments of the invention, as shown in FIG. 3,the exemplary 16APSK bit-to-symbol mapping circuit utilizes four bits(b_(4i), b_(4i+1), b_(4i+2), b_(4i+3)) each iteration and maps them intoan I value and a Q value, with i=0, 1, 2, . . . . The bit mapping logicis shown in FIG. 4. According to various embodiments of the invention,the mappings of bits are defined by:

$\left( {{I(i)},{Q(i)}} \right) = \left\{ \begin{matrix}{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,0,0} \right)} \\{\left( {{R_{1}{\sin \left( {\pi/4} \right)}},{{- R_{1}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,0,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/4} \right)}},{{- R_{2}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,1,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,0,0} \right)} \\{\left( {{R_{1}{\sin \left( {\pi/4} \right)}},{R_{1}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,0,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/4} \right)}},{R_{2}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,1,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,0,0} \right)} \\{\left( {{{- R_{1}}{\sin \left( {\pi/4} \right)}},{{- R_{1}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,0,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/4} \right)}},{{- R_{2}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,1,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,0,0} \right)} \\{\left( {{{- R_{1}}{\sin \left( {\pi/4} \right)}},{R_{1}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,0,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/4} \right)}},{R_{2}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,1,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,1,1} \right)}\end{matrix} \right.$

According to various embodiments of the invention, the bit mappingscheme of FIG. 4 uses gray mapping, meaning the binary representationsof adjacent symbols differ by only one bit. Density evolution analysisshows that given an WPC coded 16APSK system, the exemplary gray mappingscheme can provide the best threshold. The bit mapping scheme of FIG. 4also arranges bits in an order based on the values of a log likelihoodratio from the communications channel. This arrangement simplifies thedesign of interleaving scheme for 16APSK system.

Although the invention has been described by the way of exemplaryembodiments, it is to be understood that various other adaptations andmodifications may be made within the spirit and scope of the invention.Therefore, it is the object of the appended claims to cover all suchvariations and modifications as come within the true spirit and scope ofthe invention.

1. A method of digital mapping in a 16APSK system, the methodcomprising: transmitting a digital signal from a transmitter; andreceiving the digital signal at a receiver; wherein the digital signalis bit-mapped prior to the transmitting according to the followingformula, wherein R₁ is a radius of an inner ring and R₂ is a radius ofan outer ring:$\left( {{I(i)},{Q(i)}} \right) = \left\{ \begin{matrix}{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,0,0} \right)} \\{\left( {{R_{1}{\sin \left( {\pi/4} \right)}},{{- R_{1}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,0,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/4} \right)}},{{- R_{2}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,1,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,0,0} \right)} \\{\left( {{R_{1}{\sin \left( {\pi/4} \right)}},{R_{1}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,0,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/4} \right)}},{R_{2}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,1,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,0,0} \right)} \\{\left( {{{- R_{1}}{\sin \left( {\pi/4} \right)}},{{- R_{1}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,0,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/4} \right)}},{{- R_{2}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,1,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,0,0} \right)} \\{\left( {{{- R_{1}}{\sin \left( {\pi/4} \right)}},{R_{1}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,0,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/4} \right)}},{R_{2}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,1,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,1,1} \right)}\end{matrix} \right.$
 2. The method of claim 1, wherein the systemutilizes an FEC code.
 3. A digital communication system, comprising: atransmitter to transmit a digital signal; wherein the digital signalutilizes a 16APSK system with FEC coding, and the signal is bit-mappedusing gray mapping, and bits of the digital signal are ordered based onthe values of a log likelihood ratio from a communications channel. 4.The method of claim 3, wherein the FEC code is regular LDPC code.
 5. Themethod of claim 3, wherein the FEC code is irregular LDPC code.
 6. Themethod of claim 3, wherein the FEC code is regular repeat-accumulatecode.
 7. The method of claim 3, wherein the FEC code is irregularrepeat-accumulate code.
 8. A digital communication system, comprising: atransmitter to transmit a digital signal, wherein the transmittermodulates at least one mapping group having four bits (b_(4i), b_(4i+1),b_(4i+2), b_(4i+3)), for i=0, 1, 2, . . . , to a 16APSK symbol based onformula: $\left( {{I(i)},{Q(i)}} \right) = \left\{ \begin{matrix}{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,0,0} \right)} \\{\left( {{R_{1}{\sin \left( {\pi/4} \right)}},{{- R_{1}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,0,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/4} \right)}},{{- R_{2}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,1,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,0,0} \right)} \\{\left( {{R_{1}{\sin \left( {\pi/4} \right)}},{R_{1}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,0,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/4} \right)}},{R_{2}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,1,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,0,0} \right)} \\{\left( {{{- R_{1}}{\sin \left( {\pi/4} \right)}},{{- R_{1}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,0,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/4} \right)}},{{- R_{2}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,1,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,0,0} \right)} \\{\left( {{{- R_{1}}{\sin \left( {\pi/4} \right)}},{R_{1}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,0,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/4} \right)}},{R_{2}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,1,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,1,1} \right)}\end{matrix} \right.$ where R₁ is a radius of an inner ring and R₃ is aradius of an outer ring.
 9. A digital communication system, comprising:a receiver to receive a digital signal, wherein the receiver comprises ademodulator to map 16APSK symbols to estimating messages of groups offour bits (b_(4i), b_(4i+1, b) _(4i+2), b_(4i+3)), for i=0, 1, 2, . . ., based on a 16APSK constellation specification as follows:$\left( {{I(i)},{Q(i)}} \right) = \left\{ \begin{matrix}{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,0,0} \right)} \\{\left( {{R_{1}{\sin \left( {\pi/4} \right)}},{{- R_{1}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,0,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/4} \right)}},{{- R_{2}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,1,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,0,0} \right)} \\{\left( {{R_{1}{\sin \left( {\pi/4} \right)}},{R_{1}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,0,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/4} \right)}},{R_{2}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,1,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,0,0} \right)} \\{\left( {{{- R_{1}}{\sin \left( {\pi/4} \right)}},{{- R_{1}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,0,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/4} \right)}},{{- R_{2}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,1,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,0,0} \right)} \\{\left( {{{- R_{1}}{\sin \left( {\pi/4} \right)}},{R_{1}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,0,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/4} \right)}},{R_{2}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,1,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,1,1} \right)}\end{matrix} \right.$ where R₁ is a radius of an inner ring and R₂ is aradius of an outer ring.
 10. A computer readable medium to store acomputer program in which a 16APSK modulation maps groups of four bits(b_(4i), b_(4i+1), b_(4i+2), b_(4i+3)), for i=0, 1, 2, . . . , to 16APSKsymbols based on formula:$\left( {{I(i)},{Q(i)}} \right) = \left\{ \begin{matrix}{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,0,0} \right)} \\{\left( {{R_{1}{\sin \left( {\pi/4} \right)}},{{- R_{1}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,0,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/4} \right)}},{{- R_{2}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,1,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,0,1,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,0,0} \right)} \\{\left( {{R_{1}{\sin \left( {\pi/4} \right)}},{R_{1}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,0,1} \right)} \\{\left( {{R_{2}{\sin \left( {\pi/4} \right)}},{R_{2}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,1,0} \right)} \\{\left( {{R_{2}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {0,1,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{{- R_{2}}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,0,0} \right)} \\{\left( {{{- R_{1}}{\sin \left( {\pi/4} \right)}},{{- R_{1}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,0,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/4} \right)}},{{- R_{2}}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,1,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{{- R_{2}}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,0,1,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/12} \right)}},{R_{2}{\cos \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,0,0} \right)} \\{\left( {{{- R_{1}}{\sin \left( {\pi/4} \right)}},{R_{1}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,0,1} \right)} \\{\left( {{{- R_{2}}{\sin \left( {\pi/4} \right)}},{R_{2}{\cos \left( {\pi/4} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,1,0} \right)} \\{\left( {{{- R_{2}}{\cos \left( {\pi/12} \right)}},{R_{2}{\sin \left( {\pi/12} \right)}}} \right),} & {\left( {b_{4i},b_{{4i} + 1},b_{{4i} + 2},b_{{4i} + 3}} \right) = \left( {1,1,1,1} \right)}\end{matrix} \right.$ where R₁ is a radius of an inner ring and R₂ is aradius of an outer ring.